2018年1月11日

# Sums of gcd-sum functions with weights concerning the Gamma function and Bernoulli polynomials

• Isao Kiuchi
• ,

In this paper, we establish the following two identities involving the Gamma<br />
function and Bernoulli polynomials, namely $$\sum_{k\leq x}\frac{1}{k^s}<br /> \sum_{j=1}^{k^s}\log\Gamma\left(\frac{j}{k^s}\right)<br /> \sum_{\substack{d|k d^{s}|j } }(f*\mu)(d)\quad {\rm and }\quad \sum_{k\leq<br /> x}\frac{1}{k^s}\sum_{j=0}^{k^{s}-1}<br /> B_{m}\sum_{\substack{d|k d^{s}|j } } (f*\mu)(d)$$ with any fixed integer $s&gt;<br /> 1$ and any arithmetical function $f$. We give asymptotic formulas for the above<br />
with various multiplicative functions $f$. We also consider several formulas of<br />
Dirichlet series having coefficients $\gcd$-sum functions with weights<br />
concerning the Gamma function and Bernoulli polynomials.

リンク情報
arXiv
http://arxiv.org/abs/arXiv:1801.03653
URL
http://arxiv.org/abs/1801.03653v1
ID情報
• arXiv ID : arXiv:1801.03653

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